Calculus Question

Question

A 10 m ladder leans against a vertical wall and the bottom of the ladder slides away from the wall at a rate of 0.6m/sec. 

(a)  How fast is the top of the ladder sliding down the wall when the bottom of the ladder is 4 m from the wall?

1. The height of the top of the ladder is changing at a rate of _____ m/sec. This means that the top of the ladder is sliding down at a rate of ______  m/sec.

(b)  How fast is the top of the ladder sliding down the wall when the bottom of the ladder is 8 m from the wall?

1. The height of the top of the ladder is changing at a rate of _____  m/sec. The top of the ladder is sliding down at a rate of _____ m/sec.

c)  What do you notice from the previous answers?

1. The top slides down slower and slower as the bottom moves away at a constant speed.

1. The top slides down at at constant speed as the bottom moves away at a constant speed.

1. The top slides down faster and faster as the bottom moves away at a constant speed.
2. 
   

Answer 1

x is the distance measure from the bottom of the ladder to the wall.

y is the distance measure from the ground to the top of the ladder.

pythagorean says:

x^2 + y^2 = 100

2x dx + 2y dy = 0 <--- power rule

x dx + y dy = 0 <--- divides by 2

6x + y dy = 0 <---- given

(a) find dy when x = 4

4^2 + y^2 = 100

y^2 = 100 - 16 = 84

y = sqrt(84)

6(4) + sqrt(84) * dy = 0

24 + sqrt(84) * dy = 0

dy = -24/ sqrt(84)

= -24 * sqrt(84) / 84

= -6*sqrt(84)/ 21

= -2 * sqrt(84)/ 7

(b) find dy when x=8

8^2 + y^2 = 100

y^2 = 100 - 64 = 36

y = 6

6x + y dy = 0 <---- given

6(8) + 6*dy = 0

dy = -48/6 = -8

(c) the top slides down faster